According to the FAQ from Avalon Hill you must declare your moves before the units are strafed and the move must be legal assuming no loses to strafing.
We have always played that the city has to be clear of the Axis at the begining and end of the turn. Moving a unit in during the turn doesn’t change control. Control is in step five of combat. The Axis move is after the Allies move. The Axis would move into the Allied controlled city. Combat would occur and if there are no Axis units left control would stay with the Allies. That isn’t in the FAQ.
Hope that helps.